direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D21, C35⋊2S3, C15⋊3D7, C105⋊3C2, C21⋊1C10, C7⋊(C5×S3), C3⋊(C5×D7), SmallGroup(210,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C5×D21 |
Generators and relations for C5×D21
G = < a,b,c | a5=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D21
►On 105 points
Generators in S105
(1 102 74 46 36)(2 103 75 47 37)(3 104 76 48 38)(4 105 77 49 39)(5 85 78 50 40)(6 86 79 51 41)(7 87 80 52 42)(8 88 81 53 22)(9 89 82 54 23)(10 90 83 55 24)(11 91 84 56 25)(12 92 64 57 26)(13 93 65 58 27)(14 94 66 59 28)(15 95 67 60 29)(16 96 68 61 30)(17 97 69 62 31)(18 98 70 63 32)(19 99 71 43 33)(20 100 72 44 34)(21 101 73 45 35)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21)(22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)
(1 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(43 48)(44 47)(45 46)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(98 105)(99 104)(100 103)(101 102)
G:=sub<Sym(105)| (1,102,74,46,36)(2,103,75,47,37)(3,104,76,48,38)(4,105,77,49,39)(5,85,78,50,40)(6,86,79,51,41)(7,87,80,52,42)(8,88,81,53,22)(9,89,82,54,23)(10,90,83,55,24)(11,91,84,56,25)(12,92,64,57,26)(13,93,65,58,27)(14,94,66,59,28)(15,95,67,60,29)(16,96,68,61,30)(17,97,69,62,31)(18,98,70,63,32)(19,99,71,43,33)(20,100,72,44,34)(21,101,73,45,35), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,48)(44,47)(45,46)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(98,105)(99,104)(100,103)(101,102)>;
G:=Group( (1,102,74,46,36)(2,103,75,47,37)(3,104,76,48,38)(4,105,77,49,39)(5,85,78,50,40)(6,86,79,51,41)(7,87,80,52,42)(8,88,81,53,22)(9,89,82,54,23)(10,90,83,55,24)(11,91,84,56,25)(12,92,64,57,26)(13,93,65,58,27)(14,94,66,59,28)(15,95,67,60,29)(16,96,68,61,30)(17,97,69,62,31)(18,98,70,63,32)(19,99,71,43,33)(20,100,72,44,34)(21,101,73,45,35), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,48)(44,47)(45,46)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(98,105)(99,104)(100,103)(101,102) );
G=PermutationGroup([[(1,102,74,46,36),(2,103,75,47,37),(3,104,76,48,38),(4,105,77,49,39),(5,85,78,50,40),(6,86,79,51,41),(7,87,80,52,42),(8,88,81,53,22),(9,89,82,54,23),(10,90,83,55,24),(11,91,84,56,25),(12,92,64,57,26),(13,93,65,58,27),(14,94,66,59,28),(15,95,67,60,29),(16,96,68,61,30),(17,97,69,62,31),(18,98,70,63,32),(19,99,71,43,33),(20,100,72,44,34),(21,101,73,45,35)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21),(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,28),(23,27),(24,26),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(43,48),(44,47),(45,46),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(98,105),(99,104),(100,103),(101,102)]])
C5×D21 is a maximal subgroup of
C5×S3×D7 D15⋊D7
60 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 35A | ··· | 35L | 105A | ··· | 105X |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 105 | ··· | 105 |
| size | 1 | 21 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 21 | 21 | 21 | 21 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C5 | C10 | S3 | D7 | C5×S3 | D21 | C5×D7 | C5×D21 |
| kernel | C5×D21 | C105 | D21 | C21 | C35 | C15 | C7 | C5 | C3 | C1 |
| # reps | 1 | 1 | 4 | 4 | 1 | 3 | 4 | 6 | 12 | 24 |
Matrix representation of C5×D21 ►in GL2(𝔽41) generated by
| 16 | 0 |
| 0 | 16 |
| 40 | 15 |
| 6 | 32 |
| 32 | 27 |
| 35 | 9 |
G:=sub<GL(2,GF(41))| [16,0,0,16],[40,6,15,32],[32,35,27,9] >;
C5×D21 in GAP, Magma, Sage, TeX
C_5\times D_{21} % in TeX
G:=Group("C5xD21"); // GroupNames label
G:=SmallGroup(210,9);
// by ID
G=gap.SmallGroup(210,9);
# by ID
G:=PCGroup([4,-2,-5,-3,-7,242,2883]);
// Polycyclic
G:=Group<a,b,c|a^5=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export